Upper Decay Estimates for Non-Degenerate
Kirchhoff Type Dissipative Wave Equations
By Kosuke Ono
Department of Mathematical Sciences, Tokushima University, Tokushima 770-8502, JAPAN
e-mail : [email protected]
(Received September 30, 2019)
Abstract
We study on the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations ρu′′+ a(∥A1/2u(t)∥2)Au + u′= 0
and (u(0), u′(0)) = (u0, u1), where u0̸= 0 and the nonlocal
nonlin-ear term a(M ) = 1 + Mγ with γ > 0. Under the suitably smallness condition, we derive the upper decay estimates of the solution u(t) for the case of 0 < γ < 1 in addition to γ≥ 1.
2010 Mathematics Subject Classification. 35B40, 35L15
1
Introduction
Let H be a real Hilbert space with inner product (·, ·) and norm ∥ · ∥.
In this paper we investigate on the upper decay estimates of the solution
u(t) for the non-degenerate Kirchhoff type dissipative wave equations :
{
ρu′′+ a(∥A1/2u(t)
∥2)Au + u′ = 0 , t≥ 0
(u(0), u′(0)) = (u0, u1)∈ D(A) × D(A1/2) ,
(1.1)
where u = u(t) is an unknown real value function, ρ is a positive constant,
′= d/dt, A is a linear operator on H with dense domainD(A).
We assume that the operator A is self-adjoint and nonnegative such that (Av, v)≥ 0 for v ∈ D(A). The α-th power of A with dense domain D(Aα) is
denoted by Aα for α > 0, and the graph-norm of Aα is denoted by
∥v∥α =
(
∥v∥2+
∥Aαv
∥2)12 for v
∈ D(Aα). We use that
∥A1/2v
∥2 = (Av, v) for v
∈ D(A1/2).
For the non-local nonlinear term a(M ) ∈ C0([0,
∞)) ∩ C2((0,
∞)), we
as-sume that as follows :
Hyp.1 K1≤ a(M) ≤ K2+ K3Mγ for M ≥ 0 Hyp.2 0≤ a′(M )M ≤ K 4a(M ) for M > 0 Hyp.3 a′(M )M +|a′′(M )|M2≤ K 5Mγ for M > 0 with γ > 0 and Kj > 0 (j = 1, 2, 3, 4, 5).
From Hyp.1, we see that
K1M ≤ ∫ M 0 a(µ) dµ≤ ( K2+ K3 γ + 1M γ)M . (1.2)
For typical examples, we have that
a(M ) = 1 + Mγ with γ > 0 .
When the dimension is one, (1.1) describes small amplitude vibrations of an elastic string (see [3], [6]).
We denote the energy E(t) for (1.1) by
E(t) = ρ∥u′(t)∥2+∫
M (t)
0
a(µ) dµ with M (t) =∥A1/2u(t)
∥2. (1.3)
By fundamental calculation, we have the energy identity
d dtE(t) + 2∥u ′(t)∥2= 0 (1.4) and E(t) + 2 ∫ t 0 ∥u ′(s)∥2ds = E(0) (1.5) with E(0) = ρ∥u1∥2+ 2 ∫ ∥A1/2 u0∥2 0 a(µ) dµ .
Moreover, we introduce the quantities G(0) and B(0) on the initial data (u0, u1) : G(0) = ∥Au0∥ 2 ∥A1/2u 0∥2 + ρ∥A 1/2u 0∥2∥A1/2u1∥2− |(A1/2u0, A1/2u1)| a(∥A1/2u 0∥2)∥A1/2u0∥4 and B(0) = max{ ∥u1∥ 2 ∥A1/2u 0∥2 , 1 + K4 K4 (K2+ K3(K1−1E(0)) γ)2G(0) } .
In the previous paper [12], we have proved the following the global existence theorem (see [1], [2], [9], [13] for local solutions).
Theorem 1.1 Suppose that Hyp.1 and Hyp.2 are fulfilled. If the initial data (u0, u1) belong toD(A) × D(A1/2) and u0̸= 0, and moreover, the coefficient ρ
and the initial data (u0, u1) satisfy
2ρG(0)12B(0) 1
2 < 1
K4+ 1
,
then the problem (1.1) admits a unique global solution u(t) in the class C0([0,∞); D(A)) ∩ C1([0,
∞); D(A1/2))
∩ C2([0,
∞); H) and the solution u(t) satisfies
∥u(t)∥2≤ C(∥u0∥2+ E(0)) , (1.6)
K1M (t)≤ E(t) ≤ E(0) , (1.7) ρ|M′(t)| M (t) ≤ 1 K4+ 1 , (1.8) ∥Au(t)∥2 M (t) ≤ G(0) , ∥u′(t)∥2 M (t) ≤ B(0) , (1.9)
and M (t)≥ Ce−αt with some α > 0 for t≥ 0.
We do not need the assumption that γ ≥ 1 in our argument (see [4] for γ ≥ 1 that is, a(·) ∈ C1([0,∞)), and a′(M )≥ K
0 > 0 for γ > 0 (see [11] for
a(M ) = (1 + M )γ with γ > 0).
The purpose of this paper to derive upper decay estimates of the solution
u(t) of (1.1) for the case of 0 < γ < 1 in addition to γ ≥ 1, under Hyp.1, Hyp.2,
Hyp.3.
Our main result is as follows.
Theorem 1.2 Suppose that the assumption of Theorem 1.1 and Hyp.3 are
fulfilled. Then, the solution u(t) of (1.1) satisfies ∥A1/2u(t) ∥2 ≤ C(1 + t)−1, ∥u′(t)∥2+∥Au(t)∥2≤ { C(1 + t)−(1+2γ) if 0 < γ <1 2, C(1 + t)−2 if γ≥ 1 2, ∥A1/2u′(t)∥2+∥u′′(t)∥2≤ { C(1 + t)−(1+γ)(1+2γ) if 0 < γ < 12, C(1 + t)−3 if γ≥ 1 2 for t≥ 0.
The proof of Theorem 1.2 will be given by Propositions 2.2–2.5 in the next section.
The notations we use in the paper are standard. Positive constants will be denoted by C and will change from line to line.
Hyp.1 K1≤ a(M) ≤ K2+ K3Mγ for M ≥ 0 Hyp.2 0≤ a′(M )M ≤ K 4a(M ) for M > 0 Hyp.3 a′(M )M +|a′′(M )|M2≤ K 5Mγ for M > 0 with γ > 0 and Kj > 0 (j = 1, 2, 3, 4, 5).
From Hyp.1, we see that
K1M ≤ ∫ M 0 a(µ) dµ≤ ( K2+ K3 γ + 1M γ)M . (1.2)
For typical examples, we have that
a(M ) = 1 + Mγ with γ > 0 .
When the dimension is one, (1.1) describes small amplitude vibrations of an elastic string (see [3], [6]).
We denote the energy E(t) for (1.1) by
E(t) = ρ∥u′(t)∥2+∫
M (t)
0
a(µ) dµ with M (t) =∥A1/2u(t)
∥2. (1.3)
By fundamental calculation, we have the energy identity
d dtE(t) + 2∥u ′(t)∥2= 0 (1.4) and E(t) + 2 ∫ t 0 ∥u ′(s)∥2ds = E(0) (1.5) with E(0) = ρ∥u1∥2+ 2 ∫ ∥A1/2 u0∥2 0 a(µ) dµ .
Moreover, we introduce the quantities G(0) and B(0) on the initial data (u0, u1) : G(0) = ∥Au0∥ 2 ∥A1/2u 0∥2 + ρ∥A 1/2u 0∥2∥A1/2u1∥2− |(A1/2u0, A1/2u1)| a(∥A1/2u 0∥2)∥A1/2u0∥4 and B(0) = max{ ∥u1∥ 2 ∥A1/2u 0∥2 , 1 + K4 K4 (K2+ K3(K1−1E(0)) γ)2G(0) } .
In the previous paper [12], we have proved the following the global existence theorem (see [1], [2], [9], [13] for local solutions).
Theorem 1.1 Suppose that Hyp.1 and Hyp.2 are fulfilled. If the initial data (u0, u1) belong toD(A) × D(A1/2) and u0̸= 0, and moreover, the coefficient ρ
and the initial data (u0, u1) satisfy
2ρG(0)12B(0) 1
2 < 1
K4+ 1
,
then the problem (1.1) admits a unique global solution u(t) in the class C0([0,∞); D(A)) ∩ C1([0,
∞); D(A1/2))
∩ C2([0,
∞); H) and the solution u(t) satisfies
∥u(t)∥2≤ C(∥u0∥2+ E(0)) , (1.6)
K1M (t)≤ E(t) ≤ E(0) , (1.7) ρ|M′(t)| M (t) ≤ 1 K4+ 1 , (1.8) ∥Au(t)∥2 M (t) ≤ G(0) , ∥u′(t)∥2 M (t) ≤ B(0) , (1.9)
and M (t)≥ Ce−αt with some α > 0 for t≥ 0.
We do not need the assumption that γ ≥ 1 in our argument (see [4] for γ≥ 1 that is, a(·) ∈ C1([0,∞)), and a′(M )≥ K
0> 0 for γ > 0 (see [11] for
a(M ) = (1 + M )γ with γ > 0).
The purpose of this paper to derive upper decay estimates of the solution
u(t) of (1.1) for the case of 0 < γ < 1 in addition to γ≥ 1, under Hyp.1, Hyp.2,
Hyp.3.
Our main result is as follows.
Theorem 1.2 Suppose that the assumption of Theorem 1.1 and Hyp.3 are
fulfilled. Then, the solution u(t) of (1.1) satisfies ∥A1/2u(t) ∥2 ≤ C(1 + t)−1, ∥u′(t)∥2+∥Au(t)∥2≤ { C(1 + t)−(1+2γ) if 0 < γ <1 2, C(1 + t)−2 if γ≥ 1 2, ∥A1/2u′(t)∥2+∥u′′(t)∥2≤ { C(1 + t)−(1+γ)(1+2γ) if 0 < γ <12, C(1 + t)−3 if γ≥ 1 2 for t≥ 0.
The proof of Theorem 1.2 will be given by Propositions 2.2–2.5 in the next section.
The notations we use in the paper are standard. Positive constants will be denoted by C and will change from line to line.
2
Decay Rates
The following generalized Nakao type inequality is useful to derive decay estimates of energies (see [5], [7], [8], [10] for the proof).
Lemma 2.1 Let ϕ(t) be a non-negative function on [0,∞) and satisfy sup
t≤s≤t+1
ϕ(s)1+α≤ (k0ϕ(t)α+ k1(1 + t)−β)(ϕ(t)− ϕ(t + 1)) + k2(1 + t)−γ
with certain constants k0, k1, k2 ≥ 0, α > 0, β > −1, and γ > 0. Then, the
function ϕ(t) satisfies ϕ(t)≤ C0(1 + t)−θ, θ = min{ 1 + β α , γ 1 + α}
for t≥ 0 with some constant C0 depending on ϕ(0).
Using Lemma 2.1, we obtain the following energy decay for the energy E(t). Proposition 2.2 Under the assumption of Theorem 1.1, the energy E(t)
sat-isfies
E(t) = ρ∥u′(t)∥2+ ∫ M (t)
0
a(µ) dµ≤ C(1 + t)−1, (2.1)
and the solution u(t) satisfies
∥A1/2u(t)∥2+∥Au(t)∥2+∥A1/2u′(t)∥2+∥u′′(t)∥2≤ C(1 + t)−1 (2.2)
for t≥ 0.
Proof. Integrating (1.4) over [t, t + 1], we have 2
∫ t+1
t ∥u
′(s)∥2ds = E(t)
− E(t + 1) (≡ 2D(t)2) . (2.3)
Then there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥u′(t
j)∥2≤ 4D(t)2 for j = 1, 2 . (2.4)
On the other hand, taking the inner product of (1.1) with u(t), we have
a(M (t))M (t) = ρ ( ∥u′(t)∥2 −dtd(u′(t), u(t)) ) − (u′(t), u(t)) . (2.5)
Integrating (2.5) over [t1, t2], we have that
∫ t2 t1 a(M (s))M (s) ds ≤ ρ ∫ t+1 t ∥u ′(s)∥2ds + ρ 2 ∑ j=1 ∥u′(t j)∥∥u(tj)∥ + ∫ t+1 t ∥u ′(s)∥∥u(s)∥ ds
and from (2.3), (2.4), and Hyp.1 that
K1 ∫ t2 t1 M (s) ds≤ ρD(t)2+ CD(t) sup t≤s≤t+1g(s) with g(t) 2= ∥u(t)∥2, (2.6)
and from (1.2), (1.3), (1.7), (2.3), (2.6) that ∫ t2 t1 E(s) ds≤ ρ ∫ t+1 t ∥u ′(s)∥2ds +∫ t2 t1 ( K2+ K3 γ + 1M (s) γ)M (s) ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) . (2.7)
Integrating (2.3) over [t, t2], we have (2.3) and (2.7) that
E(t) = E(t2) + 2 ∫ t2 t ∥u ′(s)∥2ds ≤ 2 ∫ t2 t1 E(s) ds + ∫ t+1 t ∥u ′(s)∥2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1g(s) .
Since it holds that 2D(t)2= E(t)
− E(t + 1) ≤ E(t) by (2.3), we observe E(t)2≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2 ≤ C ( E(t) + sup t≤s≤t+1 g(s)2 ) (E(t)− E(t + 1)) . (2.8) Thus, using E(t)≤ E(0) and g(t) = ∥u(t)∥2
≤ C by (1.6) and (1.7), we have E(t)2
≤ C(E(t) − E(t + 1)) , (2.9)
and hence, applying Lemma 2.1 to (2.9), we obtain (2.1).
Moreover, we obtain that M (t)≤ K1−1E(t)≤ C(1+t)−1by (1.7),∥Au(t)∥2+
∥u′(t)∥2
≤ CM(t) ≤ C(1 + t)−1 by (2.4), and furthermore,∥u′′(t)∥2
≤ C(1 + t)−1 by (1.1), that is, the desired estimate (2.2) holds true. □
2
Decay Rates
The following generalized Nakao type inequality is useful to derive decay estimates of energies (see [5], [7], [8], [10] for the proof).
Lemma 2.1 Let ϕ(t) be a non-negative function on [0,∞) and satisfy sup
t≤s≤t+1
ϕ(s)1+α≤ (k0ϕ(t)α+ k1(1 + t)−β)(ϕ(t)− ϕ(t + 1)) + k2(1 + t)−γ
with certain constants k0, k1, k2 ≥ 0, α > 0, β > −1, and γ > 0. Then, the
function ϕ(t) satisfies ϕ(t)≤ C0(1 + t)−θ, θ = min{ 1 + β α , γ 1 + α}
for t≥ 0 with some constant C0 depending on ϕ(0).
Using Lemma 2.1, we obtain the following energy decay for the energy E(t). Proposition 2.2 Under the assumption of Theorem 1.1, the energy E(t)
sat-isfies
E(t) = ρ∥u′(t)∥2+ ∫ M (t)
0
a(µ) dµ≤ C(1 + t)−1, (2.1)
and the solution u(t) satisfies
∥A1/2u(t)∥2+∥Au(t)∥2+∥A1/2u′(t)∥2+∥u′′(t)∥2≤ C(1 + t)−1 (2.2)
for t≥ 0.
Proof. Integrating (1.4) over [t, t + 1], we have 2
∫ t+1
t ∥u
′(s)∥2ds = E(t)
− E(t + 1) (≡ 2D(t)2) . (2.3)
Then there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥u′(t
j)∥2≤ 4D(t)2 for j = 1, 2 . (2.4)
On the other hand, taking the inner product of (1.1) with u(t), we have
a(M (t))M (t) = ρ ( ∥u′(t)∥2 −dtd(u′(t), u(t)) ) − (u′(t), u(t)) . (2.5)
Integrating (2.5) over [t1, t2], we have that
∫ t2 t1 a(M (s))M (s) ds ≤ ρ ∫ t+1 t ∥u ′(s)∥2ds + ρ 2 ∑ j=1 ∥u′(t j)∥∥u(tj)∥ + ∫ t+1 t ∥u ′(s)∥∥u(s)∥ ds
and from (2.3), (2.4), and Hyp.1 that
K1 ∫ t2 t1 M (s) ds≤ ρD(t)2+ CD(t) sup t≤s≤t+1g(s) with g(t) 2= ∥u(t)∥2, (2.6)
and from (1.2), (1.3), (1.7), (2.3), (2.6) that ∫ t2 t1 E(s) ds≤ ρ ∫ t+1 t ∥u ′(s)∥2ds +∫ t2 t1 ( K2+ K3 γ + 1M (s) γ)M (s) ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) . (2.7)
Integrating (2.3) over [t, t2], we have (2.3) and (2.7) that
E(t) = E(t2) + 2 ∫ t2 t ∥u ′(s)∥2ds ≤ 2 ∫ t2 t1 E(s) ds + ∫ t+1 t ∥u ′(s)∥2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1g(s) .
Since it holds that 2D(t)2= E(t)
− E(t + 1) ≤ E(t) by (2.3), we observe E(t)2≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2 ≤ C ( E(t) + sup t≤s≤t+1 g(s)2 ) (E(t)− E(t + 1)) . (2.8) Thus, using E(t)≤ E(0) and g(t) = ∥u(t)∥2
≤ C by (1.6) and (1.7), we have E(t)2
≤ C(E(t) − E(t + 1)) , (2.9)
and hence, applying Lemma 2.1 to (2.9), we obtain (2.1).
Moreover, we obtain that M (t)≤ K1−1E(t)≤ C(1+t)−1by (1.7),∥Au(t)∥2+
∥u′(t)∥2
≤ CM(t) ≤ C(1 + t)−1 by (2.4), and furthermore,∥u′′(t)∥2
≤ C(1 + t)−1 by (1.1), that is, the desired estimate (2.2) holds true. □
Proposition 2.3 Under the assumption of Theorem 1.2, it holds that
F (t)≡ ρ∥A1/2u′(t)∥2+ a(M (t))
∥Au(t)∥2
≤ C(1 + t)−ω for t≥ 0 (2.10)
with ω = min{2 , 1 + 2γ}.
Proof. Taking the inner product of (1.1) with 2Au′(t), we have that
d dtF (t) + 2∥A 1/2u′(t)∥2= a′(M (t))M′(t)∥Au(t)∥2 (2.11) ≤ CM(t)γ+12∥Au(t)∥ 2 M (t) ∥A 1/2u′(t)∥
and from the Young inequality that
d
dtF (t) +∥A
1/2u′(t)∥2
≤ Cf(t)2 with f (t)2= M (t)2γ+1∥Au(t)∥4
M (t)2 . (2.12)
Integrating (2.12) over [t, t + 1], we have ∫ t+1 t ∥A 1/2u′(s)∥2ds = F (t) − F (t + 1) + C sup t≤s≤t+1 f (s)2 ( ≡ D(t)2) . (2.13)
Then, there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥A1/2u′(tj)∥2≤ 4D(t)2 for j = 1, 2 . (2.14)
On the other hand, taking the inner product of (1.1) with Au(t), we have
a(M (t))∥Au(t)∥2= ρ ( ∥A1/2u′(t)∥2 −dtd(A1/2u′, A1/2u) ) − (A1/2u′, A1/2u) and hence F (t) = 2ρ∥A1/2u′(t)∥2− ρd dt(A 1/2u′, A1/2u) − (A1/2u′, A1/2u) . (2.15) Integrating (2.15) over [t1, t2], we have from (2.13) and (2.14) that
∫ t2 t1 F (s) ds ≤ 2ρ ∫ t+1 t ∥A 1/2u′(s)∥2ds + ρ 2 ∑ j=1 ∥A1/2u′(t j)∥∥A1/2u(tj)∥ + ∫ t+1 t ∥A 1/2u′(s)∥∥A1/2u(s) ∥ ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) with g(t)2= M (t) . (2.16)
Moreover, there exists t∗∈ [t1, t2] such that
F (t∗)≤ 2
∫ t2 t1
F (s) ds . (2.17)
For τ ∈ [t, t + 1], integrating (2.11) over [τ, t∗] (or [t∗, τ ]), we have from
(2.12) and (2.17) that F (τ ) = F (t∗) + ∫ t∗ τ ( 2∥A1/2u′(s)∥2− a′(M (s))M′(s)∥Au(s)∥2)ds ≤ 2 ∫ t2 t1 F (s) ds + C ∫ t+1 t ∥A 1/2u′(s)∥2ds + C∫ t+1 t f (s)2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) + C sup t≤s≤t+1 f (s)2.
Since it holds that
D(t)2= F (t)− F (t + 1) + C sup t≤s≤t+1f (s) 2 ≤ F (t) + sup t≤s≤t+1f (s) 2 by (2.13), we observe sup t≤s≤t+1F (s) 2 ≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2+ C sup t≤s≤t+1 f (s)4 ≤ C ( F (t) + sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) (F (t)− F (t + 1)) + CF (t) sup t≤s≤t+1f (s) 2+ C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2 and hence sup t≤s≤t+1 F (s)2 ≤ C ( F (t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (F (t)− F (t + 1)) + C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2. (2.18)
Since it holds that
f (t)2= M (t)2γ+1∥Au(t)∥4 M (t)2 ≤ CM(t) 2γ+1 ≤ C(1 + t)−(1+2γ) M (t)2γ∥Au(t)∥ 2 M (t) ∥Au(t)∥ 2 ≤ CM(t)2γ ∥Au(t)∥2 ≤ C(1 + t)−2γF (t)
Proposition 2.3 Under the assumption of Theorem 1.2, it holds that
F (t)≡ ρ∥A1/2u′(t)∥2+ a(M (t))
∥Au(t)∥2
≤ C(1 + t)−ω for t≥ 0 (2.10)
with ω = min{2 , 1 + 2γ}.
Proof. Taking the inner product of (1.1) with 2Au′(t), we have that
d dtF (t) + 2∥A 1/2u′(t)∥2= a′(M (t))M′(t)∥Au(t)∥2 (2.11) ≤ CM(t)γ+12∥Au(t)∥ 2 M (t) ∥A 1/2u′(t)∥
and from the Young inequality that
d
dtF (t) +∥A
1/2u′(t)∥2
≤ Cf(t)2 with f (t)2= M (t)2γ+1∥Au(t)∥4
M (t)2 . (2.12)
Integrating (2.12) over [t, t + 1], we have ∫ t+1 t ∥A 1/2u′(s)∥2ds = F (t) − F (t + 1) + C sup t≤s≤t+1 f (s)2 ( ≡ D(t)2) . (2.13)
Then, there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥A1/2u′(tj)∥2≤ 4D(t)2 for j = 1, 2 . (2.14)
On the other hand, taking the inner product of (1.1) with Au(t), we have
a(M (t))∥Au(t)∥2= ρ ( ∥A1/2u′(t)∥2 −dtd(A1/2u′, A1/2u) ) − (A1/2u′, A1/2u) and hence F (t) = 2ρ∥A1/2u′(t)∥2− ρd dt(A 1/2u′, A1/2u) − (A1/2u′, A1/2u) . (2.15) Integrating (2.15) over [t1, t2], we have from (2.13) and (2.14) that
∫ t2 t1 F (s) ds ≤ 2ρ ∫ t+1 t ∥A 1/2u′(s)∥2ds + ρ 2 ∑ j=1 ∥A1/2u′(t j)∥∥A1/2u(tj)∥ + ∫ t+1 t ∥A 1/2u′(s)∥∥A1/2u(s) ∥ ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) with g(t)2= M (t) . (2.16)
Moreover, there exists t∗∈ [t1, t2] such that
F (t∗)≤ 2
∫ t2 t1
F (s) ds . (2.17)
For τ ∈ [t, t + 1], integrating (2.11) over [τ, t∗] (or [t∗, τ ]), we have from
(2.12) and (2.17) that F (τ ) = F (t∗) + ∫ t∗ τ ( 2∥A1/2u′(s)∥2− a′(M (s))M′(s)∥Au(s)∥2)ds ≤ 2 ∫ t2 t1 F (s) ds + C ∫ t+1 t ∥A 1/2u′(s)∥2ds + C∫ t+1 t f (s)2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) + C sup t≤s≤t+1 f (s)2.
Since it holds that
D(t)2= F (t)− F (t + 1) + C sup t≤s≤t+1f (s) 2 ≤ F (t) + sup t≤s≤t+1f (s) 2 by (2.13), we observe sup t≤s≤t+1F (s) 2 ≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2+ C sup t≤s≤t+1 f (s)4 ≤ C ( F (t) + sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) (F (t)− F (t + 1)) + CF (t) sup t≤s≤t+1f (s) 2+ C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2 and hence sup t≤s≤t+1 F (s)2 ≤ C ( F (t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (F (t)− F (t + 1)) + C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2. (2.18)
Since it holds that
f (t)2= M (t)2γ+1∥Au(t)∥4 M (t)2 ≤ CM(t) 2γ+1 ≤ C(1 + t)−(1+2γ) M (t)2γ∥Au(t)∥ 2 M (t) ∥Au(t)∥ 2 ≤ CM(t)2γ ∥Au(t)∥2 ≤ C(1 + t)−2γF (t)
and g(t)2= M (t) ≤ C(1 + t)−1, we have sup t≤s≤t+1 F (s)2≤ C(F (t) + (1 + t)−1)(F (t)− F (t + 1)) + C(1 + t)−(1+2γ) sup t≤s≤t+1 F (s) and hence sup t≤s≤t+1 F (s)2≤ C(F (t) + (1 + t)−1)(F (t)− F (t + 1)) + C(1 + t)−2(1+2γ). (2.19)
Thus, applying Lemma 2.1 to (2.19), we obtain
F (t)≤ C(1 + t)−ω with ω = min{2 , 1 + 2γ} which implies the desired estimate (2.10). □
Proposition 2.4 Under the assumption of Theorem 1.2, it holds that
∥u′(t)∥ ≤ C(1 + t)−ω for t≥ 0 (2.20)
with ω = min{2 , 1 + 2γ}.
Proof. Taking the inner product of (1.1) with 2u′(t), we have
ρd dt∥u
′(t)∥2+ 2
∥u′(t)∥2=−2a(M(t))(Au(t), u′(t)) , and by the Young inequality we observe
ρd dt∥u ′(t)∥2+ ∥u′(t)∥2 ≤ a(M(t))2 ∥Au(t)∥2.
Thus, from (1.7) and (2.10) we drive the desired estimate (2.20). □ Proposition 2.5 Under the assumption of Theorem 1.2, it holds that
L(t)≡ ρ∥u′′(t)∥2+ a(M (t)) ∥A1/2u′(t)∥2+a′(M (t)) 2 |M ′(t)|2 ≤ C(1 + t)−σ for t≥ 0 (2.21) with σ = min{3 , (1 + γ)(1 + 2γ)}.
Proof. Taking the inner product of (1.1) differentiated with respect to t with 2u′′(t), we have d dtL(t) + 2∥u ′′(t)∥2 = 3a′(M (t))M′(t)∥A1/2u′(t)∥2+a′′(M (t)) 2 (M ′(t))3 (2.22) ≤ Cf(t)2 with f (t)2= M (t)γ|M′(t)| M (t) ∥A 1/2u′(t)∥2. (2.23)
Integrating (2.23) over [t, t + 1], we have 2 ∫ t+1 t ∥u ′′(s)∥2ds ≤ L(t) − L(t + 1) + C sup t≤s≤t+1f (s) 2 ( ≡ 2D(t)2) . (2.24)
Then, there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥u′′(tj)∥2≤ 4D(t)2 for j = 1, 2 . (2.25)
On the other hand, taking the inner product of (1.1) differentiated with respect to t with u′(t), we have
a(M (t))∥A1/2u′(t)∥2+a′(M (t)) 2 |M ′(t)|2 = ρ ( ∥u′′(t)∥2− d dt(u ′′(t), u′(t)))− (u′′(t), u′(t)) and hence L(t) = 2ρ∥u′′(t)∥2 − ρdtd(u′′(t), u′(t))− (u′′(t), u′(t)) . (2.26) Integrating (2.26) over [t1, t2], we observe from (2.24) and (2.25) that
∫ t2 t1 L(s) ds ≤ 2ρ ∫ t+1 t ∥u ′′(s)∥2ds + ρ 2 ∑ j=1 ∥u′′(tj)∥∥u′(tj)∥ + ∫ t+1 t ∥u ′′(s)∥∥u′(s)∥ ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) with g(t)2=∥u′(t)∥2. (2.27) Moreover, there exists t∗∈ [t1, t2] such that
L(t∗)≤ 2 ∫ t2
t1
L(s) ds . (2.28)
For τ ∈ [t, t + 1], integrating (2.22) over [τ, t∗] (or [t∗, τ ]), we have from
(2.23) and (2.28) that L(τ ) = L(t∗) + ∫ t∗ τ ( 2ρ∥u′′(s)∥2 − 3a′(M (t))M′(s)∥A1/2u′(s)∥2+a(M (s)) 2 (M ′(s))3 ) ds ≤ 2 ∫ t2 t1 L(s) ds + C ∫ t+1 t ∥u ′′(s)∥2ds + C∫ t+1 t f (s)2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) + C sup t≤s≤t+1 f (s)2.
and g(t)2= M (t) ≤ C(1 + t)−1, we have sup t≤s≤t+1 F (s)2≤ C(F (t) + (1 + t)−1)(F (t)− F (t + 1)) + C(1 + t)−(1+2γ) sup t≤s≤t+1 F (s) and hence sup t≤s≤t+1 F (s)2≤ C(F (t) + (1 + t)−1)(F (t)− F (t + 1)) + C(1 + t)−2(1+2γ). (2.19)
Thus, applying Lemma 2.1 to (2.19), we obtain
F (t)≤ C(1 + t)−ω with ω = min{2 , 1 + 2γ} which implies the desired estimate (2.10). □
Proposition 2.4 Under the assumption of Theorem 1.2, it holds that
∥u′(t)∥ ≤ C(1 + t)−ω for t≥ 0 (2.20)
with ω = min{2 , 1 + 2γ}.
Proof. Taking the inner product of (1.1) with 2u′(t), we have
ρd dt∥u
′(t)∥2+ 2
∥u′(t)∥2=−2a(M(t))(Au(t), u′(t)) , and by the Young inequality we observe
ρd dt∥u ′(t)∥2+ ∥u′(t)∥2 ≤ a(M(t))2 ∥Au(t)∥2.
Thus, from (1.7) and (2.10) we drive the desired estimate (2.20). □ Proposition 2.5 Under the assumption of Theorem 1.2, it holds that
L(t)≡ ρ∥u′′(t)∥2+ a(M (t)) ∥A1/2u′(t)∥2+a′(M (t)) 2 |M ′(t)|2 ≤ C(1 + t)−σ for t≥ 0 (2.21) with σ = min{3 , (1 + γ)(1 + 2γ)}.
Proof. Taking the inner product of (1.1) differentiated with respect to t with 2u′′(t), we have d dtL(t) + 2∥u ′′(t)∥2 = 3a′(M (t))M′(t)∥A1/2u′(t)∥2+a′′(M (t)) 2 (M ′(t))3 (2.22) ≤ Cf(t)2 with f (t)2= M (t)γ|M′(t)| M (t) ∥A 1/2u′(t)∥2. (2.23)
Integrating (2.23) over [t, t + 1], we have 2 ∫ t+1 t ∥u ′′(s)∥2ds ≤ L(t) − L(t + 1) + C sup t≤s≤t+1f (s) 2 ( ≡ 2D(t)2) . (2.24)
Then, there exist two numbers t1∈ [t, t + 1/4] and t2∈ [t + 3/4, t + 1] such that
∥u′′(tj)∥2≤ 4D(t)2 for j = 1, 2 . (2.25)
On the other hand, taking the inner product of (1.1) differentiated with respect to t with u′(t), we have
a(M (t))∥A1/2u′(t)∥2+a′(M (t)) 2 |M ′(t)|2 = ρ ( ∥u′′(t)∥2− d dt(u ′′(t), u′(t)))− (u′′(t), u′(t)) and hence L(t) = 2ρ∥u′′(t)∥2 − ρdtd(u′′(t), u′(t))− (u′′(t), u′(t)) . (2.26) Integrating (2.26) over [t1, t2], we observe from (2.24) and (2.25) that
∫ t2 t1 L(s) ds ≤ 2ρ ∫ t+1 t ∥u ′′(s)∥2ds + ρ 2 ∑ j=1 ∥u′′(tj)∥∥u′(tj)∥ + ∫ t+1 t ∥u ′′(s)∥∥u′(s)∥ ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) with g(t)2=∥u′(t)∥2. (2.27) Moreover, there exists t∗∈ [t1, t2] such that
L(t∗)≤ 2 ∫ t2
t1
L(s) ds . (2.28)
For τ ∈ [t, t + 1], integrating (2.22) over [τ, t∗] (or [t∗, τ ]), we have from
(2.23) and (2.28) that L(τ ) = L(t∗) + ∫ t∗ τ ( 2ρ∥u′′(s)∥2 − 3a′(M (t))M′(s)∥A1/2u′(s)∥2+a(M (s)) 2 (M ′(s))3 ) ds ≤ 2 ∫ t2 t1 L(s) ds + C ∫ t+1 t ∥u ′′(s)∥2ds + C∫ t+1 t f (s)2ds ≤ CD(t)2+ CD(t) sup t≤s≤t+1 g(s) + C sup t≤s≤t+1 f (s)2.
Since it holds that D(t)2= L(t) − L(t + 1) + C sup t≤s≤t+1 f (s)2 ≤ L(t) + sup t≤s≤t+1 f (s)2 by (2.24), we observe sup t≤s≤t+1L(s) 2 ≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2+ C sup t≤s≤t+1 f (s)4 ≤ C ( L(t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (L(t)− L(t + 1)) + CL(t) sup t≤s≤t+1f (s) 2+ C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2 and hence sup t≤s≤t+1 L(s)2 ≤ C ( L(t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (L(t)− L(t + 1)) + C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2. (2.29)
(i) When 0 < γ < 12, we put ω = 1 + 2γ. Since it holds that
f (t)2≤ 2∥Au(t)∥ M (t)12 ∥u′(t)∥1−2γ M (t)12 ∥u ′(t)∥2γ ∥A1/2u′(t)∥2 ≤ C∥u′(t)∥2γ∥A1/2u′(t)∥2≤ { C(1 + t)−(1+γ)ω C(1 + t)−γωL(t) and g(t)2= ∥u′(t)∥2 ≤ C(1 + t)−ω, we have sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−(1+γ)ω sup t≤s≤t+1 L(s) and hence sup t≤s≤t+1L(t) 2 ≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−2(1+γ)ω. (2.30)
Thus, applying Lemma 2.1 to (2.30), we obtain
L(t)≤ C(1 + t)−σ with σ ={ω + 1 , (1 + γ)ω} = (1 + γ)(1 + 2γ)
which implies the desired estimate (2.21) for 0 < γ < 1 2.
(ii) When γ≥ 1
2, we put ω = 2. Since it holds that
f (t)2≤ 2M(t)γ−12∥Au(t)∥ M (t)12 ∥u ′(t)∥∥A1/2u′(t)∥ ≤ CM(t)γ−12∥u′(t)∥∥A1/2u′(t)∥ ≤ { C(1 + t)−(γ+3ω−1 2 ) C(1 + t)−(γ+ω−12 )L(t) and g(t)2= ∥u′(t)∥2 ≤ C(1 + t)−ω, we have sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−(γ+3γ2−1) sup t≤s≤t+1 L(s) and hence sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−2(γ+3γ2−1). (2.31)
Thus, applying Lemma 2.1 to (2.31), we obtain
L(t)≤ C(1 + t)−σ with σ ={ω + 1 , γ + 3γ− 1 2 } = 3 which implies the desired estimate (2.21) for γ≥ 1
2. □
Proof of Theorem 1.2. Gathering Propositions 2.2–2.5, we conclude Theorem
1.2. □
References
[1] A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Math. Methods Appl. Sci. 14 (1991) 177–195.
[2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305–330.
[3] G.F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945) 157–165.
[4] M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates, J. Differen-tial Equations 245 (2008) 2979–3007.
Since it holds that D(t)2= L(t) − L(t + 1) + C sup t≤s≤t+1 f (s)2 ≤ L(t) + sup t≤s≤t+1 f (s)2 by (2.24), we observe sup t≤s≤t+1L(s) 2 ≤ C ( D(t)2+ sup t≤s≤t+1 g(s)2 ) D(t)2+ C sup t≤s≤t+1 f (s)4 ≤ C ( L(t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (L(t)− L(t + 1)) + CL(t) sup t≤s≤t+1f (s) 2+ C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2 and hence sup t≤s≤t+1 L(s)2 ≤ C ( L(t) + sup t≤s≤t+1 f (s)2+ sup t≤s≤t+1 g(s)2 ) (L(t)− L(t + 1)) + C ( sup t≤s≤t+1f (s) 2+ sup t≤s≤t+1g(s) 2 ) sup t≤s≤t+1f (s) 2. (2.29)
(i) When 0 < γ < 12, we put ω = 1 + 2γ. Since it holds that
f (t)2≤ 2∥Au(t)∥ M (t)12 ∥u′(t)∥1−2γ M (t)12 ∥u ′(t)∥2γ ∥A1/2u′(t)∥2 ≤ C∥u′(t)∥2γ∥A1/2u′(t)∥2≤ { C(1 + t)−(1+γ)ω C(1 + t)−γωL(t) and g(t)2= ∥u′(t)∥2 ≤ C(1 + t)−ω, we have sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−(1+γ)ω sup t≤s≤t+1 L(s) and hence sup t≤s≤t+1L(t) 2 ≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−2(1+γ)ω. (2.30)
Thus, applying Lemma 2.1 to (2.30), we obtain
L(t)≤ C(1 + t)−σ with σ ={ω + 1 , (1 + γ)ω} = (1 + γ)(1 + 2γ)
which implies the desired estimate (2.21) for 0 < γ <1 2.
(ii) When γ≥ 1
2, we put ω = 2. Since it holds that
f (t)2≤ 2M(t)γ−12∥Au(t)∥ M (t)12 ∥u ′(t)∥∥A1/2u′(t)∥ ≤ CM(t)γ−12∥u′(t)∥∥A1/2u′(t)∥ ≤ { C(1 + t)−(γ+3ω−1 2 ) C(1 + t)−(γ+ω−12 )L(t) and g(t)2= ∥u′(t)∥2 ≤ C(1 + t)−ω, we have sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−(γ+3γ2−1) sup t≤s≤t+1 L(s) and hence sup t≤s≤t+1 L(t)2≤ C(L(t) + (1 + t)−ω)(L(t)− L(t + 1)) + C(1 + t)−2(γ+3γ2−1). (2.31)
Thus, applying Lemma 2.1 to (2.31), we obtain
L(t)≤ C(1 + t)−σ with σ ={ω + 1 , γ + 3γ− 1 2 } = 3 which implies the desired estimate (2.21) for γ≥ 1
2. □
Proof of Theorem 1.2. Gathering Propositions 2.2–2.5, we conclude Theorem
1.2. □
References
[1] A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Math. Methods Appl. Sci. 14 (1991) 177–195.
[2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305–330.
[3] G.F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945) 157–165.
[4] M. Ghisi and M. Gobbino, Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates, J. Differen-tial Equations 245 (2008) 2979–3007.
[5] S. Kawashima, M. Nakao, and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (1995) 617–653.
[6] G. Kirchhoff, Vorlesungen ¨uber Mechanik, Teubner, Leipzig, 1883. [7] M. Nakao, Decay of solutions of some nonlinear evolution equations, J.
Math. Anal. Appl. 60 (1977) 542–549.
[8] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 214 (1993) 325– 342.
[9] K. Ono, Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings, Funkcial. Ekvac. 40 (1997) 255–270.
[10] K. Ono, On sharp decay estimates of solutions for mildly degenerate dis-sipative wave equations of Kirchhoff type, Math. Methods Appl. Sci. 34 (2011) 1339–1352.
[11] K. Ono, Asymptotic behavior of solutions for Kirchhoff type dissipative wave equations in unbounded domains J. Math. Tokushima Univ., 61 (2017) 37–54.
[12] K. Ono, Lower decay estimates for non-degenerate Kirchhoff type dissipa-tive wave equations, J. Math. Tokushima Univ., 52 (2018) 39–52.
[13] W.A. Strauss, Nonlinear wave equations, CBMS Regional Conference Se-ries in Mathematics, Vol.73, Amer. Math. Soc., Providence, RI, 1989.
